Seismic data processing method for RMO picking

ABSTRACT

The invention relates to a method of processing seismic data, the said seismic data comprising a gather of seismic traces organized according to one or several acquisition parameters, comprising the steps of:
         a) defining an equation for an RMO curve as a combination of elementary functions of the acquisition parameter(s),   b) determining an RMO curve from the equation of step (a) as a combination of orthogonal elementary functions   c) for a given time or at a given depth, determining the coefficients of the combination that optimize the semblance of traces along the RMO curve.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is (1) based on and claims priority to FrenchApplication No. 0503882 filed on Apr. 19, 2005; and this application is(2) a continuation of U.S. application Ser. No. 11/407,743 filed on Apr.19, 2006 now U.S. Pat. No. 7,804,739, both of which applications arehereby incorporated by reference for all purposes in their entirety andare assigned to the assignee of the present invention.

STATEMENT REGARDING FEDERALLY-SPONSORED RESEARCH OR DEVELOPMENT

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REFERENCE TO SEQUENCE LISTING

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BACKGROUND OF THE INVENTION

The invention relates to the field of processing data recorded forseismic imaging purposes.

The purpose of seismic imaging is to generate high-resolution images ofthe subsoil from acoustic reflection measurements.

Conventionally, in seismic exploration, a plurality of seismic sourcesand receivers is distributed on the ground surface at a distance fromeach other. The seismic sources are activated to produce seismic wavesthat travel through the subsoil. These seismic waves undergo deviationsas they propagate. They are refracted, reflected and diffracted at thegeological interfaces of the subsoil. Certain waves that have travelledthrough the subsoil are detected by seismic receivers and are recordedas a function of time in the form of signals (called traces). Recordedsignals then have to be processed by a migration operation to obtain animage of underground geological structures. The migration operationconsists of causing reflections recorded along the correspondinginterfaces to converge.

During the processing, the stacking step consists of adding acousticreflections derived from a same point in the subsoil. This stepincreases the signal to noise ratio and the amplitude ratio betweenprimary and multiple reflections.

This is done by collecting traces into Common Image Gathers.

For example, assuming the subsoil is horizontally stratified with nolateral variation of acoustic velocities, those traces which illuminatethe same point in the subsoil for variable source-receiver distances(offsets) are those with a common midpoint between the source andreceiver.

This is why traces can be collected into gathers of Common Mid-Pointtraces.

However, waves reflected in the subsoil are recorded at arrival timesthat vary as a function of the offset. Therefore, before traces can beadded, they have to be corrected to bring them to a common reference,namely the zero offset trace. This correction is made during a so-calledNormal Move Out (NMO) correction step.

In general, it is considered that the time at which the same event isrecorded varies as a function of the offset along a hyperbolic NMO curvethat depends on the average wave propagation velocity in the subsoil.For each time at zero offset, an NMO curve is determined by successiveapproximations of the velocity and an evaluation of the semblance oftraces along the corresponding curve. The determination of NMO curvesprovides a means of correcting traces so as to align reflections on alltraces so that they can be stacked.

However, most of the time, the NMO correction is not sufficientlyprecise and distortions remain. An additional correction is made duringa so-called Residual Move Out (RMO) step.

In general, it is assumed that the residual correction is of theparabolic type.

On this subject, reference is made to the publication <<Robustestimation of dense 3D stacking velocities from automated picking>>,Franck Adler, Simon Brandwood, 69th Ann. Internat. Mtg., SEG 1999,Expanded Abstracts. The authors suggest an RMO correction defined by theequation:τ(x,t)=x ²(V ⁻² −V _(ref) ⁻²)/2t

where τ is the RMO correction, x is the offset; t is the time at zerooffset, V_(ref) is a reference velocity function and V is an updatedspeed.

However, seismic prospecting nowadays leads to the use of seismicsources and receivers at increasing distances from each other. As aresult of longer offsets, RMO curves become more and more difficult todescribe and the parabolic model has often been found unsatisfactory.

Furthermore, there is no model according to prior art that can describeRMO distortions as a function of the azimuth.

Furthermore, techniques according to prior art cannot be used to createa homogenous RMO picking in the acquisition space.

One purpose of the invention is to derive from recorded seismic data anevaluation of RMO distortions which is more precise than with prior arttechniques.

There is provided in accordance with the invention a method ofprocessing seismic data, the said seismic data comprising a gather ofseismic traces organized according to one or several acquisitionparameters, comprising the steps of:

-   -   a) defining an equation for an RMO curve as a combination of        elementary functions of the acquisition parameter(s),    -   b) determining an RMO curve from the equation of step (a) as a        combination of orthogonal elementary functions    -   c) for a given time or at a given depth, determining the        coefficients of the combination that optimize the semblance of        traces along the RMO curve.

The method according to the invention can be used to make an RMO pickingmore precisely than with processes according to prior art. The methodenables a picking of complex RMO curves and taking account of severalacquisition parameters.

Thus, the method according to the invention can be used to obtain afiner description of RMOs distortions than is possible with processesaccording to prior art.

Furthermore, the process according to the invention is applicable to anytype of gather of traces, particularly including multi-dimensionalgathers, in other words gathers of seismic traces organized according toseveral acquisition parameters.

The process according to the invention can be used to obtain amultidimensional RMO characterization.

The process according to the invention may include one of the followingcharacteristics:

the acquisition parameter(s) is (are) chosen from among the group ofparameters consisting of the offset, azimuth angle, Incidence angle,source-receiver coordinates,

-   -   step b) is done for a plurality of sampling times or depths,    -   the method includes a step consisting off normalizing the        following elementary functions,    -   the elementary functions are polynomial or trigonometric        functions,    -   seismic data include a plurality of gathers of traces, steps a)        and b) being done independently for each gather of traces,

the process includes steps consisting of determining variations ofcoefficients as a function of the acquisition parameter(s) on aplurality of gathers of traces and filtering the coefficients.

The invention also relates to a software product for processing ofseismic data, including a medium on which programming means are recordedto be read by a computer to control the computer so that it executessteps in the process preceding it.

Other characteristics and advantages will become clear after reading thefollowing description which is purely illustrative and is in no waylimitative and should be read with reference to the attached drawingsamong which:

FIG. 1 diagrammatically shows paths of seismic waves travelling betweensource-receiver pairs with the same common midpoint,

FIG. 2 diagrammatically shows a gather of traces (CIG) obtained after anNMO correction,

FIG. 3 shows an RMO curve applied to the gather of traces in FIG. 2,

FIG. 4 is a diagram showing the sequence of steps of a processing methodaccording to one embodiment of the invention.

FIG. 1, shows a source S-receiver R pair placed on the ground surface.During a seismic acquisition, the source S is activated to generate aseismic wave that travels through the subsoil. The seismic wave isreflected at an interface and reaches receiver R. Receiver R records theamplitude of the wave that it receives during time. The record (ortrace) obtained by the receiver R contains a signal corresponding to thereflection of the wave on the interface.

Note:

O is the midpoint between the source S and the receiver R,

d is the distance between the source S and the receiver R, namely theoffset,

(sr_(x),sr_(y)) are the coordinates of the R-S segment at the groundsurface, in an (O, x, y, z) coordinate system,

θ is the azimuth angle of the R-S segment in the (O, x, y, z) coordinatesystem,

α is the angle between the incident wave and the reflected wave at thereflection point (incidence angle).

It will be understood that a plurality of receivers are placed on theground surface during a seismic acquisition.

FIG. 2 shows a CIG gather of seismic traces obtained after filtering andNMO correction. The gather contains a number N of traces. The traces areorganized according to one or several acquisition parameters d₁,d₂, . .. d_(n).

For example, the traces may be grouped into a gather of traces with acommon midpoint (CMP).

The acquisition parameters considered may he chosen from among thefollowing parameters: offset d, azimuth angle θ, incidence angle α,source-receiver coordinates (sr_(x),sr_(y)) or any other relevantparameters.

FIG. 3 shows the GIG gather on which a picking of an RMO curve has beencompleted at time t₀.

The method according to the invention creates RMO curve picking in eachgather of a plurality of CIG gathers and for each sampling depth z.

According to one embodiment of the invention, the processing methodincludes the steps shown in FIG. 4.

The following steps are carried out for each CIG gather.

According to a first step 10, an equation with an RMO curve is definedas a linear combination of elementary functions.

$\begin{matrix}{{\Delta\;{z\left( \overset{\_}{d} \right)}} = {\sum\limits_{m = 1}^{M}{a_{m} \cdot {f_{m}\left( \overset{\_}{d} \right)}}}} & \lbrack 1\rbrack\end{matrix}$

where

Δz is the RMO variation,

(a₁,a₂, . . . a_(M)) are the coefficients of the linear combination,

(ƒ₁, ƒ₂, . . . ƒ_(M)) are the elementary functions,

d=(d₁,d₂ . . . d_(n)) are the acquisition parameters considered,

M is the dimension of the elementary functions base (M>2).

The dimension M of the decomposition base depends on the topologicalcomplexity of the required RMO curves. The increase in the dimension Mincreases the precision of the description of RMO distortions.

The expression of elementary functions ƒ_(m) depends on the organizationof the CIG gather.

According to a first possibility, the traces are grouped into a commonmidpoint (CMP) gather as a function of the offset d. The elementaryfunctions are suitably:ƒ_(m)( d )=d ^(m) with m=0,. . . , M   [2]

where d=d. The combination of equation (1) is thus a polynomial.

It is noted for illustration purposes that in the case of seismic databased on compressional wave arrivals (P-waves), the combination onlycomprises even exponent terms, in other words coefficients a_(m) where mis an odd number are zero.

According to a second possibility, the traces are grouped into a commonmidpoint (CMP) gather as a function of the offset d and the azimuth θ.The elementary functions can then be defined as follows:ƒ_(m) ^(p)( d )=d ₁ ^(m−p) d ₂ ^(p) with m=1, . . . ,M and p=0, . . . ,m  [3]

where d=(d₁,d₂)=(sr_(x),sr_(y)).

According to a third possibility, the traces are grouped into a commonmidpoint (CMP) gather as a function of the incidence angle α. Elementaryfunctions can then be defined as follows:ƒ_(m)( d )=tan(d)^(m) with m=1, . . . , M   [4]

where d=d=θ.

According to a second step 20, the elementary functions ƒ_(m) arenormalised according to the sampling { d _(n=1, . . . m)} in the CIGgather. Thus, the normalized elementary functions are defined asfollows:

$\begin{matrix}{f_{\{\overset{\_}{d}\}}^{S},{{m\left( {\overset{\_}{d}}_{n} \right)} = {{f_{m}\left( {\overset{\_}{d}}_{n} \right)}/\sqrt{\sum\limits_{k = 1}^{N}{f_{m}^{2}\left( {\overset{\_}{d}}_{k} \right)}}}}} & \lbrack 5\rbrack\end{matrix}$

where

ƒ_({ d}) ^(S),m is the normalised elementary function ƒ_(m),

k is a trace of the CIG gather,

N is the number of traces in the CIG gather.

This step may be expressed in matrix form as follows:F _({ d}) ^(S) =F·S{ d}  [6]whereF_({ d}) ^(S) is a matrix with dimensions N×M, defined as F _({ d})^(S)=( ƒ _({ d}) ^(S)1, . . . , ƒ _({ d}) ^(S) M),   [7]

F is a matrix with dimensions N×M defined by F=( ƒ ₁, . . . . ƒM) whereƒ _(m)=1, . . . M=(ƒ_(m)( d ₁), . . . ,ƒ_(m)( d _(N)))^(T)   [8]

S_({ d}) is a diagonal normalisation matrix with dimensions M×M definedby

$\begin{matrix}{s_{{{\{\overset{\_}{d}\}}i},{j{({\neq i})}}} = {{0\mspace{14mu}{and}\mspace{14mu} s_{{{\{\overset{\_}{d}\}}m},m}} = {I/\sqrt{\sum\limits_{k = 1}^{N}{f_{m}^{2}\left( {\overset{\_}{d}}_{k} \right)}}}}} & \lbrack 9\rbrack\end{matrix}$

The purpose of the normalisation step 20 is to define the principalsearch directions for the NMO curve independent of the samplingdistribution of acquisition parameters d₁, d₂, . . . d_(n).

According to a third step 30, the elementary functions arcorthogonalized. Thus, orthogonal elementary functions are defined{ƒ_({ d}) _(m=1, . . . ,M( d)) ⁷⁹ }. This is done by making a breakdownof the matrix (F_({ d}) ^(S))^(T)·F_({ d}) ^(S) into singular values(SVD) such that((F _({ d}) ^(S))^(T) ·F _({ d}) ^(S) =U _({ d})·Λ_({ d}) ·U _({ d})^(T)   [10]

where

U_({ d}) is a rotation matrix with dimensions M×M formed from Meigenvectors ē_({ d}m=1, . . . ,M) of the matrix (F_({ d})^(S))^(T)·F_({ d}) ^(S).

$\begin{matrix}{U_{\{\overset{\_}{d}\}} = \left( {{\overset{\_}{e}}_{{{\{\overset{\_}{d}\}}1},\ldots\mspace{14mu},}{\overset{\_}{e}}_{{\{\overset{\_}{d}\}}M}} \right)} & \lbrack 11\rbrack \\{{U_{\{\overset{\_}{d}\}}^{T} \cdot U_{\{\overset{\_}{d}\}}} = {I = \begin{pmatrix}1 & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & 1\end{pmatrix}}} & \lbrack 12\rbrack\end{matrix}$

Λ_({ d}) is a diagonal matrix with dimensions M×M containing eigenvaluesλ_({ d}m=1, . . . ,M)

$\begin{matrix}{\Lambda_{\{\overset{\_}{d}\}} = \begin{pmatrix}\lambda_{{\{\overset{\_}{d}\}}1} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & \lambda_{{\{\overset{\_}{d}\}}M}\end{pmatrix}} & \lbrack 13\rbrack\end{matrix}$

The normalisation step 20 that precedes step 30 limits the variabilityof eigenvalues related to sampling conditions { d} in the CIG gather oftraces. Regardless of the sampling conditions, we get:

$\begin{matrix}{{{\sum\limits_{i = 1}^{M}\lambda_{{\{\overset{\_}{d}\}}i}} = M},{\forall\left\{ \overset{\_}{d} \right\}}} & \lbrack 14\rbrack\end{matrix}$

The normalisation step 20 also reduces the variability of eigenvectorsrelated to sampling conditions { d} in the CIG gather of traces. For twogiven sampling conditions { d}, and { d}₂, we have:U _({ d) ₁ _(}) ^(T) ·U _({ d) ₂ _(})≈1   [15]

Starting from relations [10] and [6], we have:(F _({ d}) ^(S) ·U _({ d}))^(T)·(F _({ d}) ^(S) ·U_({ d}))=Λ_({ d})  [16](F _({ d}) ·S _({ d}) ·U _({ d}))^(T)·(F _({ d}) ·S _({ d}) ·U_({ d}))=Λ_({ d})  [17]

A matrix of RMO distortion approximations can be obtained using:F _({ d}) ^(Λ) =F _({ d}) ·S _({ d}) ·U _({ d})=( ƒ _({ d}) ^(Λ)_(1, . . . ,) ƒ _({ d}) ^(Λ) _(M))^(T)   [18]

where

$\begin{matrix}{{{F_{{\{\overset{\_}{d}\}}m}^{\Lambda}\left( \overset{\_}{d} \right)} = {{\sum\limits_{k = 1}^{M}{{{\overset{\_}{e}}_{k,m} \cdot {f_{{\{\overset{\_}{d}\}}k}^{S}\left( \overset{\_}{d} \right)}}\mspace{14mu}{with}\mspace{14mu} m}} = 1}},\ldots\mspace{14mu},M} & \lbrack 19\rbrack\end{matrix}$

By construction, we have:( ƒ _({ d}) ^(Λ) _(i))^(T· ƒ) _({ d}) ^(Λ) _(j)=0, for i≠j   [20]( ƒ _({ d}) ^(Λ) _(i))^(T· ƒ) _({ d}) ^(Λ) _(i)=λ_(i)   [21]

As a function of the decomposition base used to define the RMO curve, weobtain:Δ z=F·ā=F _({ d}) ^(S) ·ā _({ d}) ^(S) =F _({ d}) ^(Λ) ·ā _({ d})^(Λ)  [22]whereΔ z=(Δz( d ₁), . . . ,Δz( d _(N)))   [23]ā=(a ₁ , . . . ,a _(M))^(T)   [24]ā _({ d}) ^(S)=(a ₁ ^(S) , . . . ,a _(M) ^(S))^(T)   [25]ā _({ d}) ^(Λ)=(a ₁ ^(Λ) , . . . ,a _(M) ^(Λ))^(T)   [26]Hence:ā=(F _({ d}) ^(T) ·F _({ d}))⁻¹ ·F _({ d}) ^(T) ·F _({ d}) ^(Λ) ·ā_({ d}) ^(Λ) =S _({ d}) ·U _({ d}) ·ā _({ d}) ^(Λ)  [27]ā _({ d}) ^(Λ) =U _({ d}) ^(T) ·S _({ d}) ⁻¹ ·ā  [28]

According to a fourth step 40, picking of the RMO curve is completed foreach sampling depth z (or reference time). This is done by determining aseries of coefficients {a_(m=1, . . . , M) ^(Λ)} to optimize thesemblance of traces along the RMO curve.

The RMO curve is defined in the orthogonal elementary functions base by:Λ z( d )=F _({ d}) ^(Λ) ·ā _({ d}) ^(Λ)  [29]

The coefficients are determined by iteration so as to maximise thesemblance of traces along the RMO curve.

An example parameter can be used to measure semblance along the RMOcurve, as follows:

$\begin{matrix}{{S\left( a_{m = {1\mspace{14mu}\ldots\mspace{14mu} m}}^{\Lambda} \right)} = \frac{\left( {\sum\limits_{i = 1}^{N}A_{i}} \right)^{2}}{\sum\limits_{i = 1}^{N}A_{i}^{2}}} & \lbrack 30\rbrack\end{matrix}$

where A_(i) is the amplitude value of the trace i along the RMO curve.

The value of the semblance S thus determined is between 0 and 1.

Steps 10, 20, 30 and 40 are carried out Independently for each CIGgather of traces among the plurality of gathers and for each sampleddepth z.

According to a fifth step 50, a variation of coefficients{a_(m=1, . . . , M) ^(Λ)}, is determined on the set of gathers in theplurality of CIG gathers.

Due to the orthogonality of elementary functions F_({ d}m) ^(Λ)·, eachcoefficient a_(m) ^(Λ) may be filtered independently of the othercoefficients.

The reliability of the RMO curve or in an equivalent manner the signalto noise ratio associated with each a_(m=1, . . . ,M) ^(Λ) coefficientdepends on the following conditions:

a) the CIG fold, in other words the number of acquired traces, and thenoise associated with each trace,

b) the distribution of acquisition parameters d=(d₁, d₂ . . . d_(n))within the CIG gather,

c) the signal to noise ratio of the CIG gather.

In practice, conditions a), b) and c) vary from one gather of traces toanother, and from one depth to the other. Consequently, thea_({ d}m=1, . . . ,M) ^(Λ) (x, y, z) coefficients are contaminatedby-some unwanted high-frequency noise components. In order to spatiallystabilize the automatic determination of the RMO curve, scale factorsmust be regularised and filtered. This is done by firstly projecting RMOcurves on a reference sampling grid CIG with a reference samplingconfiguration { d _(R)} according to:

$\begin{matrix}{{\Delta\;{z\left( {\overset{\_}{d}}_{R} \right)}} = {\sum\limits_{m = 1}^{M}{a_{m}^{\Lambda} \cdot {f_{m}^{\Lambda}\left( {\overset{\_}{d}}_{R} \right)}}}} & \lbrack 31\rbrack\end{matrix}$

which leads to the following relation:ā _({ d) _(R) _(}) ^(Λ) =U _(R) ^(T) ·S _(R) ⁻¹ ·S·U·ā _({ d})^(Λ)  [32]

The spatial continuity on each coefficient a_({ d) _(R) _(}m) ^(Λ)(x, y,z) can then be reinforced independently, since by construction thesecoefficients are not correlated. The precise description of RMO curveshas many applications including:

-   -   updating of effective velocity models in time imaging: NMO        (Normal MoveOut), DMO (Dip MoveOut), Pre-STM (Pre-Stack Time        Migration),    -   updating of the interval velocity model in depth imaging,    -   optimum stacking of image gathers (CIG),    -   an AVO (Amplitude Versus Offset) and AVA (Amplitude Versus        Angle) analysis,    -   characterization of azimuth anisotropy.

The method can be used to obtain a description from a gather of tracesin a single pass, in other words without it being necessary to split thegather into offset slices, scatter angle slices or azimuth sectors.

This approach provides a robust and precise RMO picking method.

The RMO picking is optimized regardless of CIG gather samplingconditions.

Data management is thus facilitated. In particular, the single passapproach provides a means of reducing the steps of setting parametersfor data for processing purposes.

The process is applicable to any type of trace gather, includingmultidimensional gathers, for example such as multi-azimuth gathers. Themethod enables a multidimensional RMO picking (for exampletwo-dimensional or three-dimensional picking).

1. A method of processing seismic data comprising a gather of seismic traces organized according to one or several acquisition parameters, each seismic trace being a time dependent amplitude signal and being associated with a given position on the ground zone over which the acquisition has been carried out, the method comprising the steps of: determining with a computer a residual move-out (RMO) curve by picking in the gather of seismic traces; defining an equation for the RMO curve, as a combination of elementary functions of the acquisition parameter(s); orthogonalizing the elementary functions to determine an RMO curve as a combination of orthogonal elementary functions; for a given time or at a given depth, determining the coefficients of the combination that optimize the semblance of traces along the RMO curve; and processing the seismic traces in view of these coefficients and generating an image of underground geological structures corresponding to the gather of seismic traces.
 2. A method according to claim 1, in which the acquisition parameter(s) is (are) chosen from among the group of parameters consisting of the offset, azimuth angle, incidence angle, source-receiver coordinates.
 3. A method according to claim 1, in which the step of determining the coefficients is completed for a plurality of sampling times or depths.
 4. A method according to claim 1, in which the step of orthogonalizing includes a step of normalizing the elementary functions.
 5. A method according to claim 1, in which the elementary functions are polynomial functions.
 6. A method according to claim 1, in which the elementary functions are trigonometric functions.
 7. A method according to claim 1, wherein the seismic data include a plurality of gathers of traces, in which all the steps are carried out independently for each gather of traces.
 8. A method according to claim 7, including the steps of determining variations of the coefficients as a function of the acquisition parameter(s) on a plurality of gathers of traces and filtering the coefficients.
 9. A method of processing seismic data comprising a common mid-point gather organized in accordance with an offset (d), the method comprising the steps of: defining an equation for a residual move-out (RMO) curve as a linear combination of elementary functions of the offset of the form f_(m)(d)=d^(m), where f_(m) is an elementary function and m is a number between zero and M, with M being a dimension of the elementary functions; orthogonalizing with a computer the elementary functions to determine an RMO curve as a combination of orthogonal elementary functions; and for a given time or at a given depth, determining the coefficients of the combination that optimize the semblance of traces along the RMO curve and generating an image of underground geological structures corresponding to the common mid-point gather.
 10. A method of processing seismic data comprising a common mid-point gather organized in accordance with an offset (d), the method comprising the steps of: defining an equation for an RMO curve as a linear combination of elementary functions of the offset of the form f_(m)(d)=d^(m), where f_(m) is an elementary function and m is a number between zero and M, with M being a dimension of the elementary functions; normalizing and orthogonalizing with a computer the elementary functions to determine an RMO curve as a combination of orthogonal elementary functions; and for a given time or at a given depth, determining the coefficients of the combination that optimize the semblance of traces along the RMO curve and generating an image of underground geological structures corresponding to the common mid-point gather.
 11. A method of processing seismic data comprising a gather of seismic traces organized according to one or several acquisition parameters, each seismic trace being a time dependent amplitude signal and being associated with a given position on the ground zone over which the acquisition has been carried out, the method comprising the steps of: receiving the seismic data from a sensor; determining, based on the received seismic data from the sensor, a residual move-out (RMO) curve by picking in the gather of seismic traces; defining an equation for the RMO curve, as a combination of elementary functions of the acquisition parameter(s); orthogonalizing the elementary functions to determine an RMO curve as a combination of orthogonal elementary functions; for a given time or at a given depth, determining the coefficients of the combination that optimize the semblance of traces along the RMO curve; and processing the seismic traces in view of these coefficients curve and generating an image of underground geological structures corresponding to the gather of the seismic traces.
 12. The method according to claim 11, in which the acquisition parameter(s) is (are) chosen from among the group of parameters consisting of the offset, azimuth angle, incidence angle, source-receiver coordinates.
 13. The method according to claim 11, in which the step of determining the coefficients is completed for a plurality of sampling times or depths.
 14. The method according to claim 11, in which the step of orthogonalizing including prior to orthogonalizing the step of normalizing the elementary functions.
 15. The method according to claim 11, in which the elementary functions are polynomial functions.
 16. The method according to claim 11, in which the elementary functions are trigonometric functions.
 17. The method according to claim 11, wherein seismic data include a plurality of gathers of traces, in which the steps are carried out independently for each gather of traces.
 18. The method according to claim 17, further comprising the steps of determining variations of the coefficients as a function of the acquisition parameter(s) on a plurality of gathers of traces and filtering the coefficients.
 19. A method of processing seismic data comprising a common mid-point gather organized in accordance with an offset (d), the method comprising the steps of: receiving the seismic data from a sensor; determining the common mid-point gather based on the received seismic data; defining an equation for a residual move-out (RMO) curve as a linear combination of elementary functions of the offset of the form f_(m)(d)=d^(m), where f_(m) is an elementary function and m is a number between zero and M, with M being a dimension of the elementary functions; orthogonalizing the elementary functions to determine an RMO curve as a combination of orthogonal elementary functions; and for a given time or at a given depth, determining the coefficients of the combination that optimize the semblance of traces along the RMO curve and generating an image of underground geological structures corresponding to the common mid-point gather. 